Projekt inovace předmětů – Teorie a praxe dluhopisů

Část II

B. Stádník

University of Economics in Prague

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Empirical duration concept

Empirical duration concept Why?

Macaulay’s duration could be roughly financially interpreted as the percentage change of a bond price

after interest rate shift of 1% along the whole zero-coupon curve (along the whole spectrum of

maturities) when initial price of the bond is equaled to100% and the rate shift is 1 p.p.

In practical life Macaulay’s duration has no significant reason as the parallel curve shift is very rare thus

we have to find a certain value which respects the realistic empirical zero-coupon curves shifts, so the

result will be much closer to the reality. We name this measure as “Empirical duration”.

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Empirical duration concept, methodology

Methodology could be divided into following steps:

1. We take USD empirical zero-coupon curve rates on daily basis, in this research from USD market (figure1).

2. Based on the empirical data we calculate the daily change of price of fixed coupon bonds with maturities 1 to 30

years (equation 2b). The coupon is, for the demonstration, set to be 3% p.a.

3. Based on the daily price changes we calculate adequate daily empirical duration in the same way how the

Macaulay’s duration is calculated (equation 9). 4. We define and calculate Empirical duration as mean of daily empirical durations.

5. We calculate the values of Empirical duration for different coupons.

Empirical duration concept, basic theory

𝑃(𝑌𝑇𝑀) = 1+ 𝑌𝑇𝑀 𝑙𝑇 −1 𝑐 + 𝑐(1 + 𝑌𝑇𝑀) + 𝑐1 + 𝑌𝑇𝑀 2 + ⋯+ 𝑐 + 1001 + 𝑌𝑇𝑀 𝑛−1

𝑃(𝑌𝑇𝑀) = 𝑐1 + 𝑌𝑇𝑀 + 𝑐1+ 𝑌𝑇𝑀 2 + ⋯+ 𝑐 + 1001 + 𝑌𝑇𝑀 𝑛

here P(YTM) is the dirty price of a bond determined in the percentage of its face value on purchasing day, c is the coupon

rate per the coupon period, YTM is the yield to maturity per the coupon period, l is the number of days to the next coupon

payment, n is the number of coupon payments till the maturity and T is the number of days inside the coupon period.

in the special case when we purchase a bond on the day with zero accrued interest (could be for example an ex-

coupon day) and the clean price equals to the dirty price we can use the formula (2a) for the approximation of the

required clean price development.

Empirical duration concept, basic theory

𝑃(𝑖1,𝑖2,… 𝑖𝑚𝑎𝑡) = 𝑐1 + 𝑖1 + 𝑐1+ 𝑖2 2 +⋯+ 𝑐 + 1001 + 𝑖𝑚𝑎𝑡 𝑛

𝑓 𝑎 + ℎ = 𝑓 𝑎 + 𝑓′ 𝑎 ℎ+ 𝑓′′ 𝑎 ℎ22! + 𝑓′′′ 𝑎 ℎ33! + ⋯+ 𝑅

𝛥𝑃(𝑌𝑇𝑀) = 𝑃′ 𝑌𝑇𝑀 𝛥𝑌𝑇𝑀 + 𝑃′′ 𝑌𝑇𝑀 𝛥𝑌𝑇𝑀22! + 𝑃′′′ 𝑌𝑇𝑀 𝛥𝑌𝑇𝑀33! +⋯+ 𝑅

We may consider the total price also to be the sum of total prices of n zero-coupon bonds:

where i1, i2,… imat are zero-coupon rates for maturities 1,2,….n years.

Based on the Taylor’s theorem for a real-valued function 𝑓 differentiable at the point 𝑎, there is a polynomial

approximation of a higher degree (quadratic, cubic, quartic…) at the fixed point 𝑎. Taylor´s theorem provides this

approximation in a sufficiently small neighbourhood (h) of the fixed point a:

With the substitution: h for ∆i and f(a) for P. Consequently, it can be shown that, the expression holds:

Empirical duration concept, basic theory

𝛥𝑃 𝑌𝑇𝑀 ≅ −𝐷𝑈𝑅𝑀𝐴𝐶 𝑃 𝑌𝑇𝑀1 + 𝑌𝑇𝑀 𝛥𝑌𝑇𝑀 + 12𝑃 𝑌𝑇𝑀 𝐶𝑂𝑁𝑉𝛥𝑌𝑇𝑀2

𝐷𝑈𝑅𝑀𝐴𝐶 = 𝑘𝑐(1 + 𝑌𝑇𝑀)𝑘 + 100𝑛1 + 𝑌𝑇𝑀 𝑛𝑛𝑘=1 𝑃

The general measure of volatility and only up to the second order approximation:

Macaulay’s duration can be specified using the following shorthand notation:

Empirical duration concept, definition

𝐷𝑈𝑅𝐸𝑀𝑃𝑑(𝑚𝑎𝑡, 𝑑, 𝑌𝑇𝑀) ≅ − 𝑌𝑇𝑀𝑃𝑚𝑎𝑡,𝑑 𝛥𝑃𝑚𝑎𝑡,𝑑𝛥𝑌𝑇𝑀

DUR EMP_d (mat,d,YTM) empirical duration on daily basis as function of maturity, day, yield to maturity (figure1)

YTM yield to maturity

ΔYTM change of YTM

P (mat,d,) price of bond with respect to its time to maturity and the structure of zero coupon curve

on day d

ΔP (mat,d,) change of price between d and d-1 (according to figure 1)

d number of days (from the beginning of time series, figure1)

𝑫𝑼𝑹𝑬𝑴𝑷 = 𝑬 𝑫𝑼𝑹𝑬𝑴𝑷_𝒅(𝒎𝒂𝒕,𝒅, 𝒀𝑻𝑴)

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Empirical duration concept 𝐷𝑈𝑅𝐸𝑀𝑃_𝑃𝑂𝑅𝑇𝐹 ≅ 𝑤𝑗𝐷𝑈𝑅𝐸𝑀𝑃 𝑗𝑛

𝑗=1

Where wj is the weight of jth asset in the portfolio.

Empirical duration concept, USD rates market

USD rates

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Empirical duration concept, empirical observations

USD rates Source: Author

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Empirical duration concept, empirical observations

USD rates

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Empirical duration concept, empirical observations

USD rates

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Empirical duration concept, empirical observations

USD rates

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Empirical duration concept, empirical observations

USD rates

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Empirical duration concept, empirical observations

USD rates

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Empirical duration concept, results

USD rates

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Empirical duration concept, results

MAT 1 2 3 4 5 6 7 8 9 10

E.

DUR 1,00 1,97 2,91 3,82 4,71 5,56 6,38 7,17 7,92 8,65

MAT 11 12 13 14 15 16 17 18 19 20

E.

DUR 9,34 10,00 10,63 11,23 11,79 12,33 12,84 13,32 13,78 14,21

MAT 21 22 23 24 25 26 27 28 29 30

E.

DUR 14,61 15,00 15,36 15,70 16,02 16,33 16,61 16,88 17,14 17,38 USD rates

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Empirical duration concept, results

USD rates

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Empirical duration concept, results MAT\COUPON 1 2 3 4 5 6 7

1 1 1 1 1 1 1 1

2 1,99 1,98 1,97 1,96 1,95 1,94 1,94

3 2,97 2,94 2,91 2,89 2,86 2,84 2,82

4 3,94 3,88 3,82 3,77 3,73 3,68 3,64

5 4,89 4,79 4,71 4,63 4,55 4,49 4,42

6 5,83 5,69 5,56 5,44 5,34 5,24 5,16

7 6,76 6,56 6,38 6,22 6,08 5,96 5,85

8 7,67 7,4 7,17 6,97 6,79 6,64 6,51

9 8,57 8,22 7,92 7,68 7,47 7,28 7,12

10 9,45 9,01 8,65 8,35 8,1 7,89 7,7

11 10,3 9,77 9,34 8,99 8,7 8,46 8,25

12 11,14 10,5 10 9,6 9,28 9 8,77

13 11,96 11,2 10,63 10,18 9,81 9,52 9,26

14 12,75 11,88 11,23 10,72 10,32 10 9,73

15 13,52 12,52 11,79 11,24 10,81 10,46 10,17

16 14,27 13,13 12,33 11,73 11,26 10,89 10,59

17 14,99 13,72 12,84 12,19 11,7 11,3 10,98

18 15,69 14,28 13,32 12,63 12,1 11,69 11,36

19 16,36 14,81 13,78 13,04 12,49 12,06 11,72

20 17,01 15,31 14,21 13,43 12,86 12,41 12,06

21 17,62 15,78 14,61 13,8 13,2 12,75 12,38

22 18,22 16,23 15 14,15 13,53 13,06 12,69

23 18,78 16,66 15,36 14,48 13,85 13,37 12,99

24 19,32 17,05 15,7 14,79 14,14 13,65 13,27

25 19,84 17,43 16,02 15,09 14,43 13,93 13,54

26 20,32 17,79 16,33 15,37 14,7 14,19 13,8

27 20,79 18,12 16,61 15,64 14,95 14,45 14,05

28 21,22 18,43 16,88 15,89 15,2 14,69 14,29

29 21,64 18,73 17,14 16,13 15,43 14,92 14,52

30 22,03 19,01 17,38 16,36 15,65 15,14 14,75

USD rates

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Empirical duration concept, results

USD rates

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Empirical duration concept, EUR rates market

EUR rates

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Empirical duration concept, empirical observations

EUR rates

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Empirical duration concept, empirical observations

EUR rates

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Empirical duration concept, empirical observations

EUR rates

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Empirical duration concept, empirical observations

EUR rates

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Empirical duration concept, empirical observations

EUR rates

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Empirical duration concept, results

EUR rates

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Empirical duration concept, results

MAT. 1 2 3 4 5 6 7 8 9 10

E. DUR 1,00 2,00 2,91 3,83 4,72 5,60 6,43 7,24 8,03 8,80

MAT. 11 12 13 14 15 16 17 18 19 20

E. DUR 9,53 10,25 10,94 11,60 12,25 12,85 13,47 14,04 14,60 15,14

MAT. 21 22 23 24 25 26 27 28 29 30

E. DUR 15,66 16,15 16,64 17,10 17,55 17,98 18,39 18,79 19,18 19,55

EUR rates Source: Author

Empirical duration concept, results

EUR rates

Source: Author

Empirical duration concept, results MAT\COUPON 1 2 3 4 5 6 7

1 1,00 1,00 1,00 1,00 1,00 1,00 1,00

2 1,99 1,98 1,97 1,96 1,95 1,95 1,94

3 2,97 2,94 2,92 2,89 2,87 2,84 2,82

4 3,94 3,88 3,83 3,78 3,74 3,70 3,66

5 4,90 4,81 4,72 4,65 4,58 4,51 4,45

6 5,85 5,71 5,59 5,48 5,38 5,29 5,21

7 6,78 6,59 6,43 6,28 6,15 6,04 5,93

8 7,71 7,46 7,24 7,06 6,89 6,75 6,62

9 8,62 8,30 8,03 7,80 7,60 7,43 7,28

10 9,51 9,12 8,79 8,52 8,29 8,09 7,91

11 10,40 9,92 9,53 9,21 8,94 8,71 8,51

12 11,27 10,70 10,25 9,88 9,57 9,31 9,09

13 12,12 11,45 10,94 10,52 10,18 9,89 9,65

14 12,96 12,19 11,60 11,14 10,76 10,45 10,18

15 13,78 12,90 12,25 11,73 11,32 10,98 10,70

16 14,58 13,60 12,87 12,30 11,86 11,50 11,20

17 15,37 14,27 13,46 12,86 12,38 11,99 11,67

18 16,14 14,92 14,04 13,39 12,88 12,47 12,14

19 16,90 15,55 14,60 13,90 13,36 12,93 12,58

20 17,63 16,15 15,14 14,39 13,83 13,38 13,02

21 18,35 16,74 15,65 14,87 14,28 13,81 13,43

22 19,05 17,31 16,15 15,33 14,71 14,23 13,84

23 19,74 17,86 16,64 15,77 15,13 14,63 14,23

24 20,40 18,39 17,10 16,20 15,53 15,02 14,61

25 21,05 18,90 17,55 16,61 15,92 15,40 14,98

26 21,67 19,39 17,98 17,01 16,30 15,76 15,34

27 22,28 19,87 18,39 17,39 16,67 16,12 15,69

28 22,87 20,33 18,79 17,76 17,02 16,47 16,03

29 23,44 20,77 19,18 18,12 17,37 16,80 16,36

30 24,00 21,19 19,55 18,47 17,70 17,13 16,68

EUR rates

Source: Author

Conclusion

Long bonds are not so risky as they should be according Macaulya’s duration.

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